# Number theory to estimate lower bound of spectrum in quantum mechanics?

I recently worked on the following idea: Eigenvalue of an Euler product type operator?

## Summary of the idea

We represent numbers by infinite dimensional matrices such as $3$ will have all $0$s except the $1$st row will have a $1$ in the third column, the $2$nd row will have a $1$ in the $6$th row and the $r$th row and $3r$th column also have $1$s:

$$\hat 3 = | 1 \rangle \langle 3 | + | 2 \rangle \langle 6 | + | 3 \rangle \langle 9 | + \dots = \begin{bmatrix} 0 &0 & 1 &0 & 0& \dots & 0 \\ 0 &0 & 0 &0 &0 & 1 & \dots & \\ \vdots \\ 0 & 0 & 0 & \dots \end{bmatrix}$$

Similarly we can define $\hat 2$:

$$\hat 2 = | 1 \rangle \langle 2 | + | 2 \rangle \langle 4 | + | 3 \rangle \langle 6 | + \dots =\begin{bmatrix} 0 &1 & 0 &0 & 0& \dots & 0 \\ 0 &0 & 0 &1 &0 & 0 & \dots & \\ 0 &0 & 0 &0 &0 & 1 & \dots & \\ \vdots \end{bmatrix}$$

One notices that these numbers obey multiplication

$$\hat 6 = \hat 3 . \hat 2 = \hat 2 . \hat 3$$

where the $r$th row of the $6r$th column has a $1$

One can also use this to define an Euler like product formula:

$$\hat \zeta (s) = \hat 1 + \hat 2^s + \hat 3^s + \dots = (1- \hat 2^s)^{-1}(1- \hat 3^s)^{-1}(1- \hat 5^s)^{-1} \dots$$

Note: $\zeta(1) |\lambda \rangle = |\text{factors of } \lambda \rangle$

## My Observation

Let us define the following ladder operators $a$ and $a^\dagger$ from quantum mechanics:

$$A^\dagger|n \rangle = | n+1 \rangle$$

Now we make the following observation:

$$\frac{A^\dagger}{\hat I- A^\dagger} \geq \hat \zeta(1) - \hat I$$

Where I is the identity or $\hat 1$. To see what $\frac{A^\dagger}{1- A^\dagger}$ looks like apply it $\hat 1$ as: $\frac{A^\dagger}{1- A^\dagger} \hat 1$. In the sense:

$$\langle m |\frac{A^\dagger}{\hat I- A^\dagger} |n \rangle \geq \langle m |(\hat \zeta(1) - \hat I) |n \rangle$$

In fact for $n>1$:

$$\langle m |\frac{A^{\dagger 2}}{\hat I- A^\dagger} |n \rangle \geq \langle m |(\hat \zeta(1) - \hat I) |n -1 \rangle$$

Subtracting the above equations:

$$\langle m|A^\dagger| n \rangle \geq \langle m |(\hat \zeta(1) - \hat I) ( |n \rangle - |n -1 \rangle) \geq \langle m |(\hat \zeta(1) - \hat I) |n \rangle$$

## Question

Given a Hamiltonian can be expressed as annihilation and creation operators can the above expression be used to get some lower bounds on the difficult to compute spectrums?

$$H(a^\dagger, a) \geq H'(\hat \zeta,\hat \zeta^\dagger)$$

• Your multiplicative operators are all lowering, annihilation type operators, not raising, creation ones. Dec 26, 2018 at 21:49